Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Circulation and lift

Lift with no wing at all

A spinning cylinder has no camber, no aerofoil section and no trailing edge, and it lifts exactly as hard as its circulation says it should. Which settles what lift is caused by.
14 min read 6 figures Lift is circulationThe exact theory is wrongTaught wrongly, everywhere

Every explanation of lift should be tested against the case that has none of the usual furniture. A circular cylinder has no camber, no curved upper surface, no sharp trailing edge and no aerofoil section of any kind.

Spin it, and it lifts.

Lift from a spinning cylinderA circular cylinder with circulation round it. There is no aerofoil section, no camber and no sharp trailing edge, and it lifts — which rules out shape as the explanation and leaves circulation as the thing that matters.lift 3.393= ρUΓ = 3.400drag 1.5e-16 — still zeroideal flow — no shape, only circulationΓ = -3.4
Fig. 1 A cylinder with circulation round it, solved exactly. The streamlines are crowded above and spread below, and the force perpendicular to the stream is exactly the value the circulation predicts.

Why this settles the argument

Take the popular explanations in turn and ask what each predicts here.

The shape story — curved on top, flat underneath, so the upper path is longer — predicts nothing. A circle’s upper and lower surfaces are identical, so the path lengths are equal and the equal transit account gives exactly zero lift.

The constriction story — the wing squeezes the flow above it — predicts nothing either, and by the same symmetry.

The circulation account predicts a lift of ρUΓ\rho U \Gamma, upward if the circulation is one way and downward if it is the other, and it says nothing whatever about shape.

The cylinder lifts. So the circulation account is the one still standing, and the others are not approximations to it — they are claims about a mechanism that the cylinder demonstrably does not have.

What the spin actually does

A common misreading is that the surface “drags air round with it” and that this dragged air is the lift. That is nearly right and worth sharpening.

In a genuinely inviscid fluid a spinning cylinder would do nothing at all: with no friction there is nothing to couple the surface to the fluid, and the flow would be indifferent to whether the body was rotating. Circulation would remain zero and so would the lift.

What spin does in a real fluid is set up circulation through viscosity, in the boundary layer. The surface drags the adjacent air, the separation points shift asymmetrically — later on the side moving with the flow, earlier on the side moving against it — and the resulting flow has net circulation round the body.

Once it does, the inviscid theory takes over and gives the force accurately. So the mechanism that establishes the circulation is viscous, and the relation between circulation and lift is not. That division is the same one the Kutta condition relies on: viscosity selects, and ideal flow computes.

What the picture shows

Three features of the figure are worth pointing at, because they are the visible consequences of the circulation rather than decoration.

The streamlines are crowded above and spread below. Crowding means the same flux is passing through a narrower gap, so the flow there is faster — and by Bernoulli, at lower pressure. That asymmetry is the whole of the lift, and it exists on a shape with perfect up-down symmetry.

The stagnation points are not fore and aft. Both have slid round the body in the same direction, which is the signature the circulation leaves on the surface.

The wake is nowhere. The flow closes up perfectly behind, because this is an inviscid solution and an inviscid fluid exerts no drag on anything. A real spinning cylinder has a great deal of drag, and this figure cannot show it.

The first two are real and the third is the model’s limit. Distinguishing which features of a figure are physics and which are the model’s blind spots is most of what reading one carefully means.

More circulation, more lift

The relation is linear, which is worth checking rather than assuming.

Doubling the circulation doubles the lift exactly, because L=ρUΓL = \rho U \Gamma has no other terms in it. There is no threshold, no saturation and no shape-dependent coefficient in the ideal theory: the force is proportional to the circulation and to nothing else about the body.

That linearity is what makes the theorem so useful, and it is also the clearest indication that shape is not in the mechanism. A body’s geometry affects how much circulation a given situation produces — enormously — and once the circulation exists, the geometry has no further say in the force.

Ideal flow past a cylinder with circulation -1.5A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.lift 1.497ideal flow — inviscid, irrotational, steadycirculation -1.5
Fig. 2 Half the circulation of the figure above. The streamline asymmetry is visibly milder and the lift is exactly half, with the same cylinder in the same stream.

The same theorem, twice

The point worth dwelling on is that no new physics is required.

L=ρUΓL = \rho U \Gamma

is the same expression that gives the lift of an aerofoil. Same derivation, same assumptions, and the only thing that changed is the body it is applied to and how the circulation got there.

For a wing, the circulation is fixed by the sharp trailing edge. For a spinning cylinder, it is fixed by the rotation rate and the viscous coupling. The theorem does not care which.

That is what a good explanation looks like: one relation covering cases that appear to have nothing in common, with the differences confined to how the input is set.

A Joukowski aerofoil at 6°A cambered aerofoil in a uniform stream. The circulation is not chosen: it is whatever value makes the flow leave the sharp trailing edge smoothly, and that single condition fixes the lift.circulation round this loop: -2.445Γ = 2.445C_L = 1.212ideal flow with the Kutta condition applied6° incidence
Fig. 3 The same quantity measured round a wing. The loop, the integral and the theorem are identical to the cylinder’s; only the body inside them has changed.

What has to be true for it to work

The effect needs three things, and each is worth naming because removing any of them removes the lift.

A body that can carry circulation. Any closed body will do; it does not need to be a wing, and it does not need to be a circle either.

A mechanism to establish the circulation. For a wing, a sharp trailing edge. For a cylinder, rotation and viscosity. Without one, the circulation is zero and so is the lift.

A free stream. The theorem is ρUΓ\rho U \Gamma, and with UU zero there is no force however fast the cylinder spins. A rotating cylinder in still air experiences no Magnus lift at all — it stirs the air round itself and goes nowhere.

That third condition is the one that surprises, and it is a useful check on any proposed explanation: whatever mechanism is claimed, it has to vanish when the oncoming flow does.

Where it shows up

The effect is named for Heinrich Magnus, who studied it in 1852 while trying to explain why artillery shells drifted sideways. It is visible almost anywhere a spinning object moves through air.

A sliced tennis ball dips sharply — topspin means circulation that pushes the ball down. A struck golf ball climbs, because backspin does the opposite, and much of a golf ball’s carry is Magnus lift rather than the launch angle.

A football bent round a wall is doing the same thing about a vertical axis. Cricket’s swing bowling is a related but different mechanism, involving asymmetric boundary layer transition rather than spin, and conflating the two is a common error.

There have even been aircraft and ships using rotating cylinders instead of wings and sails — Anton Flettner built both in the 1920s. The ships work; they are periodically revived when fuel is expensive, and one or two operate commercially now. The aircraft flew and was not useful, for reasons of drag rather than lift.

What the solver computed

The figure is an exact solution, not an illustration.

lib/flow.js builds the flow as a uniform stream plus a doublet plus a vortex, all at the cylinder’s centre. The doublet makes the circle a streamline; the vortex supplies the circulation. The velocity field is closed-form.

Three things are checked before the figure renders. The velocity is tangent to the surface everywhere, to about seven parts in a thousand million million — nothing flows through the wall. The lift is computed by integrating the surface pressure, which never mentions circulation, and comes to 2.994 against the theorem’s 3.000. And the drag is computed the same way and comes to 10−1610^{-16}, which is zero: an ideal fluid exerts no drag on a spinning cylinder either.

The circulation itself is also measured, by walking a loop of radius 1.6 round the body and adding up the velocity along it. It returns −3.000001-3.000001 against the −3-3 that was put in.

The sign, and a bug worth recording

That last check exists because of an error this site made.

In the convention the solver uses, a negative circulation lifts upwards. Taking the textbook’s positive sign for the aerofoil’s Kutta circulation therefore produced a wing generating lift downwards, and every check it had passed: mass conserved, surface a streamline, measured circulation agreeing exactly with the value it had been given.

Self-consistent, and upside down.

The cylinder is what made it visible, because here the sign can be checked against something independent — the pressure integral. There is now an assertion that computes lift both ways and refuses a sign disagreement, and it is applied to every lifting figure on the site.

Ideal flow past a cylinder with circulation -3A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.lift 2.994ideal flow — inviscid, irrotational, steadycirculation -3
Fig. 4 The same cylinder drawn with the stagnation points marked. Their positions are the visible signature of the circulation’s sign: with no circulation they sit fore and aft, and circulation slides them both round the same way.

Where the stagnation points go

The clearest way to see circulation, rather than compute it, is to watch those two points.

With no circulation the flow divides at the front and rejoins at the back, and the two stagnation points sit diametrically opposite each other on the horizontal axis.

Add circulation and both slide round the cylinder in the same direction — down on one side, and the rear one up towards it. As the circulation increases they approach each other, meet at a single point on the underside, and then leave the surface entirely into the fluid.

The lift rises throughout. So the position of the stagnation points is a direct readout of how hard the cylinder is working, and the same is true of a wing: the Kutta condition is precisely a statement about where one of them sits.

Why the aeroplane failed and the ship did not

Flettner built both, and the difference between them is instructive about what lift is worth.

A rotor ship replaces sails with vertical spinning cylinders. It works: the rotors generate substantial side force from a crosswind, the drag penalty is acceptable because a ship is not trying to go fast, and the machinery is simple. Flettner’s Buckau crossed the Atlantic in 1926, and modern rotor sails are fitted to a handful of commercial vessels for fuel saving.

A rotor aeroplane replaces wings with spinning cylinders. It flies, and it is a bad aeroplane, because the lift-to-drag ratio is dreadful. A cylinder is a bluff body: it separates early and drags a wide wake, and no amount of circulation fixes that. An aerofoil gets its circulation from a shape that is also streamlined, and getting both from one object is the whole achievement of the wing.

So the Magnus effect is an excellent demonstration that circulation causes lift, and a poor way to build a lifting surface. Those are different claims, and the essay is making only the first.

The cleanest test there is

It is worth stating what makes this a good test rather than merely an interesting case.

A test of an explanation should remove everything the explanation is not about and keep everything it is. The spinning cylinder does exactly that: it removes camber, thickness distribution, surface length asymmetry, sharp edges and section shape — all the furniture the wrong explanations depend on — and keeps circulation and a free stream, which are what the right one depends on.

If lift persisted, shape was never the mechanism. It persists.

That is worth more than any amount of argument about the aerofoil, because on a wing all the candidate mechanisms are present at once and cannot be separated. The cylinder separates them.

A Joukowski aerofoil at 8°A cambered aerofoil in a uniform stream. The circulation is not chosen: it is whatever value makes the flow leave the sharp trailing edge smoothly, and that single condition fixes the lift.Γ = 2.919C_L = 1.448ideal flow with the Kutta condition applied8° incidence
Fig. 5 And the case where everything is present together. The circulation here is fixed by the trailing edge rather than by spin, and the lift follows from it by the same theorem — which is the point the cylinder was needed to establish.
Reynolds number: one number, four different flowsReynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.creepingattachedseparated, sheddingturbulentbacterium swimmingshedding begins, Re ≈ 47a thrown ballan airliner winga whaleReynolds numberinertia ÷ viscositylog₁₀ Rethe ratio decides the regime, not the size or the speed alone
Fig. 6 A caution about all of it. The ideal solution above is a good description at the right-hand end of this axis; a cylinder spinning at low Reynolds number does something quite different, and the number decides which.

What a ball actually does

A last practical note, since most encounters with this effect are sporting rather than aeronautical.

A struck golf ball leaves the clubface with backspin of several thousand revolutions per minute, and the Magnus force from it is a substantial fraction of the ball’s weight. That is why a golf ball climbs after launch rather than following a simple ballistic arc, and why loft on a club is as much about generating spin as about launch angle.

Topspin does the reverse: a heavily topspun tennis ball dips inside the baseline from a trajectory that looks certain to go long, which is exactly what makes the shot worth playing.

Sidespin bends a football about a vertical axis, and the bend increases as the ball slows because the force depends on speed while the ball’s momentum depends on it too — so the curve tightens towards the end of the flight, which is why a well-struck free kick appears to swerve late.

None of that requires anything beyond the theorem in this essay, applied about whichever axis the ball happens to be spinning.

Where the model stops

Real spinning cylinders have drag, a great deal of it, which is why Flettner’s aeroplane was not useful. Ideal flow gives none.

The lift is not unlimited. In the model, more circulation always means more lift. In reality the boundary layer eventually cannot sustain the asymmetry and the effect saturates.

The circulation is not calculable here. How much circulation a given spin rate produces is a viscous question this model cannot answer; the figure takes the circulation as given.

Two-dimensional. A real spinning ball has ends, and the flow round them is three-dimensional and messy.

Circulation has to come from somewhere

A question the steady picture leaves open, and it applies to the cylinder exactly as it does to a wing.

Kelvin’s circulation theorem says that in an inviscid fluid the circulation round any material loop is constant. Air well upstream has none. So if a body acquires circulation, an equal and opposite amount must have been created somewhere — the total cannot change.

For a wing, the balance is paid by the starting vortex: a rotating mass of air shed from the trailing edge as the wing begins to move, left behind on the runway, carrying exactly the circulation the wing took on.

For a spinning cylinder, the accounting is less tidy but the same in principle. The vorticity that ends up bound to the cylinder came from its own surface, generated by the no-slip condition acting against the rotation, and an equal and opposite amount is carried away in the wake.

In both cases the theorem is not violated; it is paid. And in both cases the payment is viscous, which is another way of saying that a genuinely inviscid fluid could never start lifting anything at all.

Two bodies, one relation

A summary worth having, because the pair of cases together says more than either alone.

a wing a spinning cylinder
shape cambered section perfect circle
sharp edge yes, essential none
what sets the circulation the Kutta condition rotation, through viscosity
upper surface longer? slightly not at all
lift ρUΓ\rho U \Gamma ρUΓ\rho U \Gamma
ideal drag zero zero

Everything in the top half of that table differs and everything in the bottom half is identical. Whatever causes lift has to live in the bottom half, and only one quantity does.

That is the argument in its shortest form, and it is why this essay exists as a separate rung rather than as a paragraph inside the main account.

The ladder from here

Nearby: the stagnation points traced as circulation varies; Flettner rotors and why they work on ships and not on aircraft; the difference between Magnus lift and cricket-ball swing; and the saturation of the effect at high spin rates.

Then back to what holds a wing up, which is the same theorem applied to a body that has a sharp edge to do its selecting for it.

Where this goes next

Γ = 2.445C_L = 1.212ideal flow with the Kutta condition applied6° incidence What actually holds a wing up Not the shape, and not the story about air meeting up again behind. A wing lifts because there is circulation round it, and the sharp trailing edge is what decides how much. the upper surface is 1.7% longerso equal transit needs the flow over it 1.7% fasterit is actually 76.5% fasterthe premise is false and the number it predicts is wrongsurface lengths and speeds measured on the solved field6° incidence The story about air meeting up again The most repeated explanation of lift says that air parting at the nose must rejoin at the tail, so the longer upper path forces a higher speed. The premise is false, and the speed it predicts is wrong by a factor of twenty. too little — the flow whips round the edgeΓ = 0.93the Kutta value — it leaves smoothlyΓ = 2.67too much — the rear stagnation point is on topΓ = 4.81ideal flow — three admissible solutions, one physical8° incidence The sharp edge decides Ideal flow round a wing admits infinitely many solutions, each with a different lift, and all of them exact. One extra requirement — that the air leaves the trailing edge instead of whipping round it — picks a single one.